A Basic Result on the Theory of Subresultants

Akritas, Alkiviadis G., Malaschonok, Gennadi I., and Vigklas, Panagiotis S.. "A Basic Result on the Theory of Subresultants." Serdica Journal of Computing 10.1 (2016): 031-048. <http://eudml.org/doc/289533>.

@article{Akritas2016,
abstract = {Given the polynomials f, g ∈ Z[x] the main result of our paper, Theorem 1, establishes a direct one-to-one correspondence between the modified Euclidean and Euclidean polynomial remainder sequences (prs’s) of f, g computed in Q[x], on one hand, and the subresultant prs of f, g computed by determinant evaluations in Z[x], on the other. An important consequence of our theorem is that the signs of Euclidean and modified Euclidean prs’s - computed either in Q[x] or in Z[x] - are uniquely determined by the corresponding signs of the subresultant prs’s. In this respect, all prs’s are uniquely "signed". Our result fills a gap in the theory of subresultant prs’s. In order to place Theorem 1 into its correct historical perspective we present a brief historical review of the subject and hint at certain aspects that need - according to our opinion - to be revised. ACM Computing Classification System (1998): F.2.1, G.1.5, I.1.2.},
author = {Akritas, Alkiviadis G., Malaschonok, Gennadi I., Vigklas, Panagiotis S.},
journal = {Serdica Journal of Computing},
keywords = {Euclidean Algorithm; Euclidean Polynomial Remainder Sequence (prs); Modified Euclidean prs; Subresultant prs; Modified Subresultant prs; Sylvester Matrices; Bezout Matrix; Sturm’s prs},
language = {eng},
number = {1},
pages = {031-048},
publisher = {Institute of Mathematics and Informatics Bulgarian Academy of Sciences},
title = {A Basic Result on the Theory of Subresultants},
url = {http://eudml.org/doc/289533},
volume = {10},
year = {2016},
}

TY - JOUR
AU - Akritas, Alkiviadis G.
AU - Malaschonok, Gennadi I.
AU - Vigklas, Panagiotis S.
TI - A Basic Result on the Theory of Subresultants
JO - Serdica Journal of Computing
PY - 2016
PB - Institute of Mathematics and Informatics Bulgarian Academy of Sciences
VL - 10
IS - 1
SP - 031
EP - 048
AB - Given the polynomials f, g ∈ Z[x] the main result of our paper, Theorem 1, establishes a direct one-to-one correspondence between the modified Euclidean and Euclidean polynomial remainder sequences (prs’s) of f, g computed in Q[x], on one hand, and the subresultant prs of f, g computed by determinant evaluations in Z[x], on the other. An important consequence of our theorem is that the signs of Euclidean and modified Euclidean prs’s - computed either in Q[x] or in Z[x] - are uniquely determined by the corresponding signs of the subresultant prs’s. In this respect, all prs’s are uniquely "signed". Our result fills a gap in the theory of subresultant prs’s. In order to place Theorem 1 into its correct historical perspective we present a brief historical review of the subject and hint at certain aspects that need - according to our opinion - to be revised. ACM Computing Classification System (1998): F.2.1, G.1.5, I.1.2.
LA - eng
KW - Euclidean Algorithm; Euclidean Polynomial Remainder Sequence (prs); Modified Euclidean prs; Subresultant prs; Modified Subresultant prs; Sylvester Matrices; Bezout Matrix; Sturm’s prs
UR - http://eudml.org/doc/289533
ER -